Solve 1/3+1/6+1/10+1/21+frac{1}{28}+frac{1}{36}+frac{1}{45}+frac{1}{55}+frac{1}{66}+frac{1}{78}+frac{1}{91}+frac{1}{105}+frac{1}{120}

$$\frac{ 1 }{ 3 } + \frac{ 1 }{ 6 } + \frac{ 1 }{ 10 } + \frac{ 1 }{ 21 } + \frac{ 1 }{ 28 } + \frac{ 1 }{ 36 } + \frac{ 1 }{ 45 } + \frac{ 1 }{ 55 } + \frac{ 1 }{ 66 } + \frac{ 1 }{ 78 } + \frac{ 1 }{ 91 } + \frac{ 1 }{ 105 } + \frac{ 1 }{ 120 }$$

Evaluate

$$\frac{97}{120}\approx 0.808333333$$

Solution Steps

Least common multiple of 3 and 6 is 6. Convert \frac{1}{3} and \frac{1}{6} to fractions with denominator 6.

$$\frac{2}{6}+\frac{1}{6}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{2}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{2+1}{6}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 2 and 1 to get 3.

$$\frac{3}{6}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{3}{6} to lowest terms by extracting and canceling out 3.

$$\frac{1}{2}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 2 and 10 is 10. Convert \frac{1}{2} and \frac{1}{10} to fractions with denominator 10.

$$\frac{5}{10}+\frac{1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{5}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$\frac{5+1}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 5 and 1 to get 6.

$$\frac{6}{10}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.

$$\frac{3}{5}+\frac{1}{21}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 5 and 21 is 105. Convert \frac{3}{5} and \frac{1}{21} to fractions with denominator 105.

$$\frac{63}{105}+\frac{5}{105}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{63}{105} and \frac{5}{105} have the same denominator, add them by adding their numerators.

$$\frac{63+5}{105}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 63 and 5 to get 68.

$$\frac{68}{105}+\frac{1}{28}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 105 and 28 is 420. Convert \frac{68}{105} and \frac{1}{28} to fractions with denominator 420.

$$\frac{272}{420}+\frac{15}{420}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{272}{420} and \frac{15}{420} have the same denominator, add them by adding their numerators.

$$\frac{272+15}{420}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 272 and 15 to get 287.

$$\frac{287}{420}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{287}{420} to lowest terms by extracting and canceling out 7.

$$\frac{41}{60}+\frac{1}{36}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 60 and 36 is 180. Convert \frac{41}{60} and \frac{1}{36} to fractions with denominator 180.

$$\frac{123}{180}+\frac{5}{180}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{123}{180} and \frac{5}{180} have the same denominator, add them by adding their numerators.

$$\frac{123+5}{180}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 123 and 5 to get 128.

$$\frac{128}{180}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{128}{180} to lowest terms by extracting and canceling out 4.

$$\frac{32}{45}+\frac{1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{32}{45} and \frac{1}{45} have the same denominator, add them by adding their numerators.

$$\frac{32+1}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 32 and 1 to get 33.

$$\frac{33}{45}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{33}{45} to lowest terms by extracting and canceling out 3.

$$\frac{11}{15}+\frac{1}{55}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 15 and 55 is 165. Convert \frac{11}{15} and \frac{1}{55} to fractions with denominator 165.

$$\frac{121}{165}+\frac{3}{165}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{121}{165} and \frac{3}{165} have the same denominator, add them by adding their numerators.

$$\frac{121+3}{165}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 121 and 3 to get 124.

$$\frac{124}{165}+\frac{1}{66}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 165 and 66 is 330. Convert \frac{124}{165} and \frac{1}{66} to fractions with denominator 330.

$$\frac{248}{330}+\frac{5}{330}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{248}{330} and \frac{5}{330} have the same denominator, add them by adding their numerators.

$$\frac{248+5}{330}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 248 and 5 to get 253.

$$\frac{253}{330}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{253}{330} to lowest terms by extracting and canceling out 11.

$$\frac{23}{30}+\frac{1}{78}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 30 and 78 is 390. Convert \frac{23}{30} and \frac{1}{78} to fractions with denominator 390.

$$\frac{299}{390}+\frac{5}{390}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{299}{390} and \frac{5}{390} have the same denominator, add them by adding their numerators.

$$\frac{299+5}{390}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Add 299 and 5 to get 304.

$$\frac{304}{390}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{304}{390} to lowest terms by extracting and canceling out 2.

$$\frac{152}{195}+\frac{1}{91}+\frac{1}{105}+\frac{1}{120}$$

Least common multiple of 195 and 91 is 1365. Convert \frac{152}{195} and \frac{1}{91} to fractions with denominator 1365.

$$\frac{1064}{1365}+\frac{15}{1365}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{1064}{1365} and \frac{15}{1365} have the same denominator, add them by adding their numerators.

$$\frac{1064+15}{1365}+\frac{1}{105}+\frac{1}{120}$$

Add 1064 and 15 to get 1079.

$$\frac{1079}{1365}+\frac{1}{105}+\frac{1}{120}$$

Reduce the fraction \frac{1079}{1365} to lowest terms by extracting and canceling out 13.

$$\frac{83}{105}+\frac{1}{105}+\frac{1}{120}$$

Since \frac{83}{105} and \frac{1}{105} have the same denominator, add them by adding their numerators.

$$\frac{83+1}{105}+\frac{1}{120}$$

Add 83 and 1 to get 84.

$$\frac{84}{105}+\frac{1}{120}$$

Reduce the fraction \frac{84}{105} to lowest terms by extracting and canceling out 21.

$$\frac{4}{5}+\frac{1}{120}$$

Least common multiple of 5 and 120 is 120. Convert \frac{4}{5} and \frac{1}{120} to fractions with denominator 120.

$$\frac{96}{120}+\frac{1}{120}$$

Since \frac{96}{120} and \frac{1}{120} have the same denominator, add them by adding their numerators.

$$\frac{96+1}{120}$$

Add 96 and 1 to get 97.

$$\frac{97}{120}$$

Factor

$$\frac{97}{2 ^ {3} \cdot 3 \cdot 5} = 0.8083333333333333$$